(square root y-3)^2=(1-square root 2y-4)^2. how do i do this???? i am clueless, pleaseeee help!!!!
i^-7. how do I do this please describe! thanks for your time and help, I really appreciate it:)
i is -1 so
i^2 = +1
i^3 = -1 = i
i^4 = +1
Note that when the exponent is odd the answer is always -1 or i.
When the exponent is even the answer is always +1, so
i^-7 = (1/i^7) = 1/i = 1/-1 = -1 = i.
Check my work.
I will answer your first question.
looks like you were solving
√(y-3) = 1 - √(2y-4) and you are squaring both sides. You will get
y-3 = 1 - 2√(2y-4) + 2y - 4
2√(2y-4) = y
let's square again
4(2y-4) = y^2
y^2 - 8y + 16 = 0
(y-4)^2 = 0
y-4 = 0
y = 4
since we squared, our answer must be verified in the original equation
LS = √y-3 = √(4-3) = 1
RS = 1 - √(2y-4) = 1 - 2 = -1
So there is no solution.
thanks that makes alot of sense:)
For the first question, let's break down the steps to solve (square root y-3)^2 = (1-square root 2y-4)^2:
Step 1: Expand the equation
- Start by expanding both sides by squaring the expressions inside the parentheses.
- On the left side, (square root y-3)^2 simplifies to (y - 3).
- On the right side, (1 - square root 2y - 4)^2 simplifies to (1 - 2√(2y - 4) + (2y - 4)).
Step 2: Simplify the equation
- Now, we have the equation (y - 3) = (1 - 2√(2y - 4) + (2y - 4)).
- Combine like terms: (y - 3) = (-3 + 2y - 2√(2y - 4)).
Step 3: Isolate the square root term
- To isolate the square root term on one side, move all other terms to the other side of the equation.
- Rearranging the equation, we have: 2√(2y - 4) = (-2y + y - 3 + 4).
- Simplifying further: 2√(2y - 4) = (-y + 1).
Step 4: Square both sides of the equation
- To eliminate the square root, square both sides of the equation.
- The equation becomes: (2√(2y - 4))^2 = (-y + 1)^2.
- Simplifying further: 4(2y - 4) = (y - 1)^2.
Step 5: Solve for y
- Expand and simplify the equation: 8y - 16 = y^2 - 2y + 1.
- Move all terms to one side: y^2 - 10y + 17 = 0.
- Solve the quadratic equation using factoring, completing the square, or the quadratic formula.
- Once you find the values of y, substitute them back into the original equation to check for extraneous solutions.
For the second question, you want to simplify i^-7.
Step 1: Understand the properties of imaginary numbers
- The imaginary unit, denoted as i, is defined as the square root of -1.
- The powers of i repeat every four exponent values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Beyond this, the pattern repeats.
Step 2: Simplify the expression
- i^-7 can be rewritten as 1 / i^7.
- Since i^4 = 1, we can simplify i^7 using the pattern mentioned above:
i^7 = i^(4+3) = i^4 * i^3 = (1) * (-i) = -i.
Step 3: Substitute the value back
- From the previous step, we found that i^7 is equal to -i. So, i^-7 is equal to 1 / (-i).
- You may choose to leave it as -(1 / i) or multiply the numerator and denominator by i to simplify it further:
-(1 / i) = -1 * (1 / i) = -i.
Therefore, i^-7 simplifies to -i.